Linear Optimal Control of Bilinear Systems with Applications to Singular Perturbations and Weak Coupling

115 Pages·1995·4.623 MB·English

by  (auth.), Zijad Aganović, Associate Professor Zoran Gajić (eds.)

Chapter 1
Introduction
In between of linear and nonlinear systems lies a very large class of so
called bilinear systems. They represent an enormous number of the real
world phenomena (Mohler, 1970, 1973, 1974, 1991; Mohler and Chen, 1970;
Mohler and Kolodziej, 1980; Bmni et al., 1974; Bahrami and Kim, 1975;
Sundareshan and Fundkowski, 1985, 1986; Williamson, 1977; Espana and
Landau, 1978). The bilinear control systems are described by the following
evolution equation
~ = A + ~_Nkuk x + Bu (1.1)
k=l
where x E ~'R is the state vector and u E ~m is the control vector. The
matrices A,B, and Nk E R~ ''×n, k = 1, .... m are of appropriate dimensions.
The product of state and control variables, that is ukx, distinguishes these
classes of systems from the linear ones, but at the same time makes bilinear
systems so general such that "every input-output map ~ can be approximated
as closely as desired by maps which arise from bilinear systems, provided
that ~ certain continuity satisfies and causality conditions" (Sussmann, 1976;
Lo, 1975). That is why A. Balakrishnan raised an important question: "Are
all nonlinear systems bilinear," (Balakrishnan, 1976).
2 NOITCUDORTNI
Very often in the literature on the bilinear control systems, the mathe-
matical model (1.1) is recorded as
= Ax + B + Mjxj u (1.2)
j=l
with Mj E ~r, .mx
A general block diagram for a bilinear control system, represented by
(1.1) is given in Figure 1.1.
u(t) f II )t(x: >
!
Figure 1.1: Block diagram for a general system bilinear control
The importance of bilinear systems has been recognized since the work
of (Wiener, 1948), who believed that they are the essence of understanding
the behavior of neural and biological computing networks. Bilinear systems
were brought to attention of control engineers due to work of R. Mohler in
the early seventies. Originally they were introduced in the study of nuclear
reactors (Mohler, 1970, 1973; Mohler and Chen, 1970; Mohler and Kolodziej,
1980), where the bilinearity of state andc ontrol variables appears naturally in
the reactor kinetic equation. Relatperdo blems such as nuclear fission, reactor
NOITCUDORTNI 3
shut down, reactor control and thermal processes are described by bilinear
dynamic equations also. During the seventies and the first part of the eighties
the bilinear control systems were the subject of very extensive research. In
the last decade they become an integral part of the modem nonlinear trend in
control and system theory so that no too many research studies dealt strictly
with the bilinear systems. However, many important results obtained for
the nonlinear control systems can be specialized and used for the bilinear
control systems.
The major importance of bilinear systems indeed lies in their applications
to the real world systems as demonstrated in the following paragraphs.
Bilinear systems naturally represent many physical processes, for example:
the basic law of mass action (Mohler, 1970, Bruni et al., 1974), dynamics of
heat exchanger with controlled flow (Bruni et al., 1974); distillation columns
(Espana and Landau, 1978); some processes in elasticity (Slemrod, 1978);
dc motor (Bruni et al., 1974); induction motor drives (Figalli et al., 1984);
mechanical brake system (Mohler, 1970, 1973); aerial combat between two
aircrafts and missile intercept problem (Wei and Pearson, 1978); modeling
and control of a small furnace, (Baheti and Mohler, 1981); control of
hydraulic rotary multi-motor systems, (Guo et al., 1994).
Many biological processes, such as the population dynamics of biological
species (Mohler, 1970; Bruni et al., 1974); water balance and.temperature
regulation in human body (Mohler, 1970, 1973, 1974); control of carbon
dioxide in lungs (Mohler, 1970); blood pressure (Mohler, 1991); immune
system (Mohler, 1991); cardiac regulator (Mohler, 1970); circulation of thy-
roxin in human body (Mohler, 1970b)e;h avior of sense organ (Bruni et al.,
1974); biological control (Williamson, 1977); growth of cancer cell popula-
tion and finding the optimal therapy (Bahrami and Kim, 1975; Sundareshan
and Fundkowski, 1985, 1986); respiratory chemostat, hormone regulation,
kidney water balance, (Mohler, 1991); dissolved oxygen process (Ko et al.,
1982); cell mass concentration in continuous culture (Yi et al., 1989) -- all
of them are described by bilinear models.
Some economic processes (a growth of a national economy (Bruni et
al., 1974), processes in ecology and socioeconomics (Mohler, 1973) may be
studied by the use of bilinear models. The bilinear systems are the adaptive
ones (Mohler, 1970). They have the variable structure strongly dependent
4 N O I T C U D O R T N I
on the control vector (1.1). Thus, the study of bilinear systems might bring
some interesting results in the field of adaptive systems as well (Ionesku and
Monopoli, 1975).
Despite vast applications of bilinear systems, they have not been studied
extensively in the domain of singular perturbations and weacko uplienxgc ept
for a few minor results (Guillen and Armada, 1980; Tzafestas and Anagnos-
tou, 1984a, 1984b; Asamoah and Jamshidi, 1987). In this book we will
pay special attention to the singularly perturbed and weakly coupled bilinear
control systems and derive techniques for decompositions of these systems
into subsystems. This simplifies implementation of the control algorithms,
speeds up real-time control and signal processing (filtering), and introduces
parallelism in the design procedures. The above features will be facilitated
by exploiting the presence of a small perturbation parameter, which in the
case of singularly perturbed systems introduces numerical ill-conditioning.
However, having obtained the system decomposition into subsystems (cor-
responding to the slow and fast time scales) the numerical ill-conditioning
is removed.
2.1 Singularly Perturbed and Weakly Coupled Bilinear
Control Systems
The application of bilinear systems is very well documented. A little bit is
known about the fact that a large number of these systemsd isplay the multi
time scale property or the singularly perturbed structure. In this section we
present some real world bilinear singularly perturbed and weakly coupled
control systems.
The theory of singular perturbations has been a rapidly developing and
highly recognized research area of control engineering in the last twenty five
years. Almost all important control aspects for linear systems have been
studied so far and valuable and practically implementable results have been
obtained. The extension of these results to nonlinears ystems happened to be
a difficult task. Only under very restrictive conditions and for very limited
classes of nonlinear systems some results were obtained (O'Malley, 1974;
INTRODUCTION 5
Chow and Kokotovic, 1978a, 1978b, 1981; Suzuki, 1981; Saberi and Khalil,
1984, 1985).
The singularly perturbed bilinearc ontrol system consistent with (1.1) is
described by the following differential equation
a'AA[=]~[ A4][~]+[B2j + ku' ]zY[
(1.3)
N,
,,,:,
where y 6 ~nl is the slow state vector, z 6 ~'~ is the fast state vector,
u 6 ~m is the control input and e is a small positive parameter. Constant
,iA ;~N
matrices i = 1,2,3,4, and B1 and B2 are of appropriate
dimensions. In the following some important bilinear systems in biological
and physical sciences that display singularly perturbed structure will be
introduced.
The neutron level control problem in a fission reactor (Mohler, 1970,
1973; Mohler and Chen, 1970) is described by the following equation
[7] = [--! --aa] [7] +u[~ ]00 [7] (1.4)
where n is neutron population, e is precursor population, u is reactivity
(it represents a control input), and a, ,3/ and ~) are known constants. It
is important to point out that A takes very small values. Typical values for
these constants clearly show the singularly perturbed structure of (1.4), (Quin,
1980), ), = 0.00001,/3 = 0.0065, and a = 0.4. Introducing a transformation
zl = n~) and x2 = c one gets
];:[]00 <,,,
z2 J -a
that is, a singularly perturbed bilinear form.
The mathematical model of a dc-motor (Bruni et al., 1974) has a bilinear
form
]:~[ = 0-'[ - - R L ]O 1:~[ ]~[lU~ O[2z~u3" (/
, 0][::]
,,6,
where l~: and x2 are the rotor current and the axis speed (state variables),
ul and u2 are the stator current and the rotor voltage (control variables), R
6 INTRODUCTION
and L are electric parameters of rotor. F and J are mechanical parameters
of the load and K is the torque constant. Since L << J is a well known fact,
the singularly perturbed structure of (1.6) is obvious.
The singularly perturbed bilinear control system consistent with form
(1.2) is represented :A[=]+:[ by
A2
A4][zY] + [BB:] u+ {[zY][MM~]} u (1.7)
with initial conditions
y(to) yo
J = z(to) [z ] °
where y E ,-'++'~ z E ~'~ are, respectively, slow and fast state variables, e
is a small positive parameter, and
1 rM~j + ~"~+zrM+jlr
{[y][M+]} --r+~l
(1.8)
Ms yIML j '[M]jJ
j=l j:nl+l
The following notation is used in order to relate (1.2) and (1.7)
A= A1 B1 Ms [y(t)
~4_cA[ - A4] B= [B_a] M= ]_~.M[ x(t)= ] (1.9,
' ' ' )t(~L
The bilinear model of induction motor drives is given by a fourth-order
differential equation (Figali et al., 1984). This frequency controlled two phase
induction motor can be put in the singularly ]lU[ perturbed ]+vr form (1.7) as given
below. The state and control variable are
] I
legs
2Y
t¢+ +
X "- Zl = ,+d. ' u = 2U -- IVqs
/-
3U L ws
2Z L sq~
where
sdC and sqC -- projections of the stator flux
sdi and sqi -- projections of the stator current
sdV and sqV -- projections of the supply voltage
ws -- slip angular frequency.
NOITCUDORTNI 7
The problem matrices have the following values
0 321.57 -0.312 0 ] -0.07]
J
-312.57 0 0 -0.312
A = , x(to) =
98.87 27059 -44.93 2.57
-27059 98.87 -2.57 -44.93 47 J
o ]o
1 -7.3 MI= 0 0 -1
B = 87.3 0 87.8 ' 0 0
87.3 -53 0 0
]i°i[ ]!°il ]!°i[
M2= 0 0 0 0 0
0 , M3= 0 ' M4= 0
0 1 0
It can be easily seen (big entries in the last two rows) that zl and 2z are
fast variables and lY and 2Y are slow variables; that is, this system displays
two time scale property also.
Many other real world biological systems either have or can be brought
in the singularly perturbed form. For example, regulation of carbon dioxide
in the respiratory system (Mohler, 1970, 1991), where time constants corre-
sponding to two time scales are determined by the lung and tissue reservoir
volumes, respectively denoted by 1V and .2V The corresponding mathemat-
ical model is given by
(1.10)
~2 = ~[c5 - c2(x~ - ~zx - c4)]
where ei, i = 1 .... ,5, are known parameters. The state variables xl and x2
respectively represent the rates of change of the lung and tissue concentration
of .2OC
8 NOITCUDORTNI
The mechanical portion of the cardiovascular system is described by the
following singularly perturbed bilinear control system (Mohler, 1991)
A1~1 = Ul(X 2 -- Xl) + lV
~2X2 = Ul(Xl -- Z2) --?)2
(1.11)
a3~3 = u2(z2 - z3) + v2
~4~4 = u2(x3 - z4) -Vl
where , Xl,X2,x;3 and 4 are compartmental X pressures in the arteries filled
from the left heart, veins to the right heart, arteries from the right heart,
and veins to the left heart, respectively; vl and 2v are cardiac outputs from
the left and right heart, respectively; )q, ,2A ,3A and 4A are corresponding
compartmental capacities (assumed to be constan0.
In addition, the mechanical brake system (Mohler, 1970), which is in
fact a system of bilinear differential equations with a huge parameter (car
mass) multiplying some derivatives (Desoer and Shena, 1970), can be put in
the singularly perturbed bilinear form. The distillation columns (Espana and
Landau, 1978) are described by three time scale bilinear models involving
huge parameters also. In (Cronin, 1987), it has been shown that the singu-
lar perturbation theory is the most efficient tool in the study of the famous
Hodgkin-Huxley model of nerve conductionl Due to very complex and non-
linear structure of this equation it is hard to believe that its linearized model
will produced satisfactory results. However, the bilinearization procedure
will considerably improve the approximation and it might result in a better
understanding of the neural conductivity.
The weakly coupled bilinear control system, in the representation con-
sistent with form (1.2), is represented by
"'
)21.1(
)[::] .o,.r
+ {[:,] )o,(,,,r
' Ly~(~o)J - Ly~ J
NOIr'CUDORTNI 9
where lY E ~nl, 2Y C ~n2, iU E ~m,, i = 1,2, and e is a small coupling
parameter, with the following notation
(1.13)
2,r+,n )aM[ jbM ]
+ Z ),~,-j(2Y
i,eM i~M
j=n, l +
where M.i E ~n, xml, ibM ~n, E ,~mx Md E ~n2xm,, iaM E t~ ~mx~n
The weakly coupled bilinear control systems are either naturally weakly
coupled or they can be obtained in the process of bilineadzation of nonlinear
weakly coupled control systems. The natural bilinear weakly coupled control
system is for example the problem of a paper making machine as given
in (Ying et al., 1992). The bilinear mathematical model of this system is
formulated according to (I.12) and (l.13) as
[
"-1.93 0 0 0 ] "1.274 1.~74 ]
493.0 -0.426 0 0
A= , B= 0 j560°-
0 0 -0.63 0 43.1
590.0 -0.103 0.413 -0.426 ll] 0
(1.14)
0 0
0 0
M, =
557.D 663.0
0 0
li]
0
= 0 ° 0 1
817.0- -0.718 '
0 0
Note that for this model only the matrix A is weakly coupled, whereas the
matrices B, M1, and 2M have no weakly coupled forms. Since in this book
we developed the linear approach to almost all bilinear-quadratic control
problems, we can use the results from (Skataric et al., 1991), in which, it
10 NOITCUDORTNI
has been shown that the classes of linear-quadratic optimal control problems
having weakly coupled system matrix and strongly coupled input matrix
(quasi weakly coupled systems) can be studied as the weakly coupled linear-
quadratic optimaclo ntrol problems by assuming the special form for the state
penalty o[matrix. T0] hi001 s can be achieved, in this particular ] example, by using
the following weighting matrices Q and R
1 0 0.09 Fo(1) o(,)
LO( ) R
Q = 0.13 0 0.1 0 = ' = (1.15)
0.09 0 0.2
Small perturbation parameter is c = 0.1.
Many other nonlinear singularly perturbed systems can be brought
into the singularly perturbed bilinear form by performing bilinearization
(Schwartz, 1988).
1.2 Book Organization
This book consists of five chapters. Chapter 1 comprises an introduction
on the general bilinear control systems and presents several examples of the
real world singularly perturbed and weakly coupled bilinear control systems.
After the introductory chapter, in the first part of this book, in Chapters 2,
and 3, we study the linear optimal control of singularly perturbed and weakly
coupled bilinear control systems. In Chapter 4 we consider new techniques
for optimization of bilinear control systems, and in Chapter 5 some future
research directions are outlined.
In Chapter 2 we study the optimization of singularly perturbed bilinear
control systems. A sequence of linear state and costate equations is con-
structed, and the open-loop solution of the optimization problem is obtainiend
terms of the reduced-order slow and fast subsystems. The ill-defined numeri-
cal problem is completely decomposed into slow and fast time scales, leading
to the reduction in the size of the requirecdo mputations and allowing parallel
processing of information. In addition, the near-optimal "closed-loop" con-
trol is obtainedi n the form of a linear approximate "feedback" control law as